Infinite Rings with Planar Zero-Divisor Graphs

Abstract

For any commutative ring R that is not a domain, there is a zero-divisor graph, denoted Γ(R), in which the vertices are the nonzero zero-divisors of R and two distinct vertices x and y are joined by an edge exactly when xy = 0. Smith (2007) characterized the graph structure of Γ(R) provided it is infinite and planar. In this article, we give a ring-theoretic characterization of R such that Γ(R) is infinite and planar.

Key Words:

• Planar
• Zero-divisor graphs

2000 Mathematics Subject Classification:

• Primary 13A99
• Secondary 05C75

ACKNOWLEDGMENTS

The author would like to thank the referee, whose careful reading and valuable comments helped improve this article.

The author was partially supported by the National Science Foundation (DMS-0700554) and by the Research Initiation Grant of Georgia State University.

Notes

Communicated by I. Swanson.